Bellringer: 1/17

• A runner is traveling 4.0m/s. Approximately, how fast is he traveling in miles per hour?

Chapter 2: Motion in One Dimension

Section 1: Displacement & Velocity

• Motion – the change of an object’s position relative to some reference point

• The study of physical concepts such as mass and force is called dynamics.

• The dynamics of objects in motion is called kinematics.

Scalar Vs. Vector

• Scalars Quantities described by a magnitude

• Vectors Quantities described by both a magnitude and

direction

Distance vs. Displacement

• Distance Scalar, how much ground an object covered during

its motion• Displacement

Vector, how far out of place an object is

• Objects that are “at rest” are not in motion.• An object undergoes displacement when its

position changes. So, displacement is the change of position.

The displacement measures how far an object has moved from it’s starting location.Note: ∆ is the Greek symbol “delta”. It means “a change in”.

xf means final position. xi means initial position.

Displacement: ∆x = xf – xi

• The displacement of an object does not always equal the distance it has traveled.

• Displacement can be either positive or negative…depending on your frame of reference.

• The right (or east) will be considered positive unless otherwise stated. Likewise with up (north).

• The left (or west) will be considered negative unless otherwise stated. Likewise with down (south).

PositiveNegative

∆x = 8 cm∆x = -4 cm

With your partner, discuss the following:1) What is this object’s position at 1 minute?2) What is the object’s position at 2 minutes?3) What is its displacement between 1 min and 5 min?4) Describe the motion of this object.

Speed Vs. Velocity

• Speed Rate at which an object covers a distance

Example: While on vacation, Lisa Carr traveled a total distance of 440 miles. Her trip took 8 hours. What was her average speed?

• Average velocity – the total displacement divided by the total amount of time during the displacement.

• Velocity may be + or - , depending on the displacement.

Velocity: v = ∆x = (xf-xi) ∆t (tf – ti)

Why refer to it as average velocity?Example:Suppose you traveled from your house to school…a

distance of 4.0 km. It took you .20 hours (12 minutes) because of heavy traffic.

Your avg velocity would be:v = 4.0 km / .20 hv = 20 km/hDid you travel at that exact speed for the entire trip?

Of course not.Because of this, velocity is an average.

Velocity and speed are not the same:Velocity requires direction.Speed refers to the numerical value of

velocity.

Avg. Speed and Avg. Velocity

• I walk 4 meters East, 2 meters South, 4 meters West, and finally 2 meters North. The entire motion lasted for 24 seconds. Determine the average speed and average velocity.

Average speed: 0.50m/sDisplacement is 0 meters, average velocity is 0m/s

Example

• A crow flies east in a straight line. It begins flying at 3:04s, and comes to a rest at 3:11s. It was flying at an

average velocity of 1.5 m/s. How far did the crow move from its initial position (in meters)?

• Using the velocity equation, we can derive another equation to find an object’s position

(Final) Displacement: xf = v∆t + xi

With your partner…1) Use the base velocity equation to derive an equation to

solve for the final displacement of an object.

Velocity can be determined using a position vs time graph.

The slope of the line corresponds to the velocity.

Position vs Time

Object 1

Object 2

Object 3

Time

Pos

ition

How could we describe the motion of these objects?

ReviewConstant VelocityPositive Velocity

Positive VelocityChanging Velocity

(acceleration)

Slow, Rightward(+)Constant Velocity

Fast, Rightward(+)Constant Velocity

Example

• Sketch a position vs. time and a velocity vs. time graph for the following: Your mother walks five meters to the kitchen at a velocity of 2 m/s, pauses by the refrigerator for three seconds, then walks back with a plate full of potato salad at a velocity of 1 m/s.

Chapter 2: Motion in One Dimension

Section 2: Accelerated Motion

• Definition: acceleration – the change in velocity over time.

• An object accelerates when it changes its motion.

–This means:»Speeding up»Slowing down»Changing direction

Acceleration: a = ∆v/∆t = vf - vi

tf - ti

• Acceleration has the derived unit of m/s2. But…what does this unit actually mean?

With your partner…1) Use the acceleration formula to show how the unit of

m/s2 is obtained.2) Explain, in plain speak, what the unit m/s2 means.

m 1

s s x =

_m_s x s

_m_s2=

m_s_

s=

–The magnitude of acceleration tells how quickly the change is happening.

–The sign tells the direction.

• Acceleration with a “+” magnitude means the object is gaining velocity in the “+” direction.

• Acceleration with a “–” magnitude means the object is slowing down (or gaining velocity in the “–” direction.

– Objects that have a constant velocity have no acceleration.

• Centripetal acceleration – the constant change of direction of an object moving in circles.

Example

• Turner’s treadmill runs with a velocity of -1.2m/s and speeds up at regular intervals during a half hour workout. After 25 min, the treadmill has a velocity of -6.5 m/s. What is the average acceleration of the treadmill during this period?

Example

• A shuttle bus slows down with an average acceleration of -1.8m/s/s. How long does it take the bus to slow from 9.0m/s to a complete stop?

Acceleration can be determined using a velocity vs time graph.

USE THE IMAGE BELOW TO:a) CALCULATE THE AVERAGE VELOCITY OF THE CAR.

b) CREATE A VELOCITY VS TIME GRAPH FOR THE CAR.

c) CALCULATE THE AVG. ACCELERATION OF THE CAR.

Velocity vs Time

The slope of a velocity vs time graph represents acceleration.

Using the base equation for acceleration, we can find other equations to find the velocity and/or displacement of moving objects that are changing their speed…

With your partner:Using the base equation for acceleration, derive two new equations to solve for initial displacement and final velocity.

Bellringer: 1/22

• What does the red line represent?

• What is the acceleration at A? B? C?

Graphing !x

t

A

B

C

A … Starts at home (origin) and goes forward slowly

B … Not moving (position remains constant as time progresses)

C … Turns around and goes in the other direction quickly, passing up home

1 – D Motion

Graphing w/ Acceleration

x

A … Start from rest south of home; increase speed gradually

B … Pass home; gradually slow to a stop (still moving north)

C … Turn around; gradually speed back up again heading south

D … Continue heading south; gradually slow to a stop near the starting point

t

A

B C

D

Tangent Lines

t

SLOPE VELOCITY

Positive Positive

Negative Negative

Zero Zero

SLOPE SPEED

Steep Fast

Gentle Slow

Flat Zero

x

On a position vs. time graph:

Increasing & Decreasing

t

x

Increasing

Decreasing

On a position vs. time graph:

Increasing means moving forward (positive direction).

Decreasing means moving backwards (negative direction).

Average Speed

What is the AVERAGE speed of the bass boat depicted in the graph?

Bass Boat Speed

0

20

50

65 65 65

85

100

125

0

20

40

60

80

100

120

140

1 2 3 4 5 6 7 8 9 10

Time (seconds)

Dis

tan

ce

(m

ete

rs)

Average Speed

Average speed is taking the total distance traveled (0 to 125 meters), and dividing by the total time (1 to 9 seconds) it takes.

Average Speed = 125 meters = 15.6 m/s 8 seconds

Bass Boat Speed

0

20

50

65 65 65

85

100

125

0

20

40

60

80

100

120

140

1 2 3 4 5 6 7 8 9 10

Time (seconds)

Dis

tan

ce

(m

ete

rs)

Instantaneous SpeedBass Boat Speed

0

20

50

65 65 65

85

100

125

0

20

40

60

80

100

120

140

1 2 3 4 5 6 7 8 9 10

Time (seconds)

Dis

tan

ce

(m

ete

rs)

What is the instantaneous speed of the bass boat at t=7 seconds?

Instantaneous SpeedBass Boat Speed

0

20

50

65 65 65

85

100

125

0

20

40

60

80

100

120

140

1 2 3 4 5 6 7 8 9 10

Time (seconds)

Dis

tan

ce

(m

ete

rs)

Instantaneous speed is speed at any given point in time. At 7 seconds, the distance is 85 meters; therefore the IS is

Instantaneous Speed = 85 meters = 12.1 m/s 7 seconds

Speed GraphsBass Boat Speed

0

20

50

65 65 65

45

15

00

10

20

30

40

50

60

70

1 2 3 4 5 6 7 8 9 10

Time (seconds)

Dis

tan

ce

(m

ete

rs)

• In what time period is the bass boat speeding up? • In what time period is the bass boat slowing down?• When is the speed NOT changing?

Graphing Speed•Speed is usually graphed using a line graph, and it depicts the distance and time.

•Time is the independent variable, and thus is ALWAYS on the x-axis.•Distance is the dependent variable, and is ALWAYS on the y-axis.

Bass Boat Speed

0

20

50

65 65 65

45

15

00

10

20

30

40

50

60

70

1 2 3 4 5 6 7 8 9 10

Time (seconds)

Dis

tanc

e (m

eter

s)

Acceleration and Velocity

• As velocity increases, so does acceleration• As velocity decreases, so does acceleration• When direction changes, so does acceleration• When there is a constant velocity, there is no

acceleration

Review

• The Kinematic Equations describe the mathematical relationships that exist between an object’s motion.–Displacement–Velocity–Acceleration–Time

• Displacement with Constant Acceleration:

• Note that:–This equation does not require acceleration.

vf + vi

2 x = t

Example

• A racing car reaches a speed of 42 m/s. It then begins a uniform negative acceleration, using its parachute and braking system, and comes to rest 5.5 s later. Find the distance that the car travels during braking. (Hint: list your givens)

Example

• A car accelerates uniformly from rest to a speed of 6.6m/s in 6.5s. Find the distance the car travels during this time.

• Velocity with Constant Acceleration:

• Note that:–This equation does not require displacement.

–∆t = (tf – ti)

vf = vi + a∆t

• Displacement with Constant Acceleration:

• Note that:–“t” is actually “∆t”.

• However, ti is usually 0.

–The equation can be rewritten to find xi.

xf = xi + vit + ½at2

Example

• A plane starting at rest at one end of a runway undergoes a uniform acceleration of 4.8m/s/s for 15s before takeoff. What is its speed at takeoff? How long must the runway be for the plane to be able to take off?

• Velocity with Constant Acceleration:

• Note that:–This equation does not require time.–∆x = (xf – xi)

vf2 = vi

2 + 2a∆x

Example

• A person pushing a stroller starts from rest, uniformly accelerating at a rate of 0.500m/s/s. What is the velocity of the stroller after it has traveled 4.75m?

Equations for Constantly Accelerated Straight-Line Motion

Bellringer 9/9

• A race car accelerates uniformly from 18.5m/s to 46.1 m/s in 2.47 seconds. Determine the acceleration of the car and the distance it traveled.

a=11.2 m/s/s and d = 79.8

Falling Objects

• free fall – downward acceleration while under the effect of gravity only.

• Gravity is a force that causes objects to accelerate downward.

–All objects free fall with an acceleration of g= -9.8 m/s2.

–gravity causes objects to speed up as they fall downward and slow as they travel upward

–we will usually neglect air resistance until FRICTION is covered in more detail.

Graphing Free Fall

• Curved line on a position vs time graph signifies an accelerated motion

Free Fall Practice Problems

• Upton Chuck is riding the Giant Drop at Great America. If Upton free falls for 2.6 seconds, what will be his final velocity and how far will he fall?

D – 33mVf = 25.5 m/s

Free Fall Practice Problems

• A feather is dropped on the moon from a height of 1.40 meters. The acceleration of gravity on the moon is 1.67 m/s/s. Determine the time for the feather to fall to the surface of the moon.

T=1.29s

Practice Problems

• A race car traveling at 44 m/s accelerates to a velocity of 22 m/s over a period of 11 seconds.

• What was the car’s displacement during this time?

360 m.

Practice Problems

• A bike rider accelerates to a velocity of 7.5 m/s during 4.5 seconds.

• If the bike had a displacement of 19 m, what was its initial velocity?

.9 m/s

Bellringer 9/10

• A tennis ball is thrown vertically upward with an initial velocity of +8.0m/s. What will the ball’s speed be when it returns to its starting point? How long will the ball take to reach its starting point?

Practice Problems

• A stone is dropped into a deep well and is heard to hit the water 3.41s after being dropped. Determine the depth of the well.

-57.0

Practice Problems

• Sketch a position time graph for the following scenarios: An object moving with a constant positive velocity An object at rest

Practice Problems

• Sketch a velocity time graph for the following motions A city bus that is moving with a constant velocity A tiger that is speeding up at a uniform rate of

acceleration while moving in the negative direction

Practice Problems

• A train accelerates from a velocity of 21 m/s with an acceleration of 3 m/s2 over a distance of 535 meters.

• What is the final velocity of the train?

60 m/s

Practice Problems

• A worker drops a wrench from the top of a tower 80.0m tall. What is the velocity when the wrench strikes the ground?

-39.6 m/s

Practice Problems

• A car moving eastward along a straight road increases its speed uniformly from 16 m/s to 32 m/s in 10.0s. What is the car’s average acceleration? What is the car’s average velocity? How far did the car move while accelerating?